ITM Web of Conferences 36, 04001 (2021) https://doi.org/10.1051/itmconf/20213604001 ICMSA 2020 An alternative approach in finding the stationary queue length distribution of a queueing system with negative customers Siew Khew Koh1*, Ching Herny Chin1, Yi Fei Tan2, and Tan Ching Ng3 1Centre for Mathematical Sciences, Universiti Tunku Abdul Rahman, Jalan Sungai Long, Bandar Sungai Long, 43000 Kajang, Selangor, Malaysia 2Multimedia University, Persiaran Multimedia, 63100 Cyberjaya, Selangor, Malaysia 3Universiti Tunku Abdul Rahman, Jalan Sungai Long, Bandar Sungai Long, 43000 Kajang, Selangor, Malaysia Abstract. A single-server queueing system with negative customers is considered in this paper. One positive customer will be removed from the head of the queue if any negative customer is present. The distribution of the interarrival time for the positive customer is assumed to have a rate that tends to a constant as time t tends to infinity. An alternative approach will be proposed to derive a set of equations to find the stationary probabilities. The stationary probabilities will then be used to find the stationary queue length distribution. Numerical examples will be presented and compared to the results found using the analytical method and simulation procedure. The advantage of using the proposed alternative approach will be discussed in this paper. 1 Introduction The development of queueing system with negative customers can be dated back to the work of Gelenbe [1, 2]. It is known as G-queue in acknowledgement of Gelenbe's introduction of the model. The simplest form of killing strategy in G-queue is removal of one positive customer at the head (RCH) or at the end (RCE) immediately by a negative customer that arrives to the queue. Various removal disciplines of a discrete-time queueing system were studied by Atencia and Moreno [3]. Since the works of Gelenbe (see [1, 2] and [4-7]), G- queue has drawn great attention from researchers to apply the concept in diverse fields, such as manufacturing system, industrial engineering, computer and communication networks. Survey on the analysis and applications of G-queue can be referred to [7-10]. In Harrison and Pitel [11], an M/M/1 G-queue was considered and the sojourn time distribution was found by using Laplace transform. They [12] extended the idea to a tandem pair G-queue and the response time distribution was computed. The expressions to find the equilibrium queue length distribution of an M/G/1 queue were also provided by them in [13]. Similar to Harrison and Pitel [11], Shin [14] derived expressions to find the sojourn time distribution by taking into consideration various killing disciplines on a multi-server G- *Corresponding author: [email protected] © The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons Attribution License 4.0 (http://creativecommons.org/licenses/by/4.0/). ITM Web of Conferences 36, 04001 (2021) https://doi.org/10.1051/itmconf/20213604001 ICMSA 2020 queue. The original concept of Gelenbe’s work was generalised by Zhu and Zhang [15] in which negative customer may also receive service when arrives to the queue. Abbas and Assani [16] investigated the performance of the system that is affected by negative customers in a GI/M/1 queue. Recently, the concept of G-queue has also been applied by Teoh [17] in modelling the framework for delay in aircraft arrival. Most of the early work on G-queue focus on continuous-time queueing system. The discrete-time counterparts have recently received increasing attention due to its application in digital communication systems and mobile network. For discrete-time G-queue, the readers may refer to [18-22]. An alternative approach will be presented in this paper to find the G-queue’s stationary queue length distribution. RCH killing discipline is considered and the interarrival time of the positive customers is assumed to have a constant asymptotic rate as time t tends to infinity. We call such distribution a CAR distribution. Similar model has been considered in [23]. However, the service time for positive customer in [23] is assumed to have a non-exponential distribution while the interarrival time remains as exponentially distributed. The same approach has been successfully applied to the models in [23-27]. The numerical result will be compared to the result computed using the analytical method presented in [24]. Analytical method could be preferred over numerical method if we want an exact result. However, numerical method is more versatile. Moreover when exponential assumption of both the interarrival and service time distributions is relaxed, the expressions derived using the analytical method may have a complicated form and will not be easy to solve. Hence, it is worthwhile to study the proposed method in this paper since simple probabilistic approach is used to derive the equations for the stationary probabilities. The values of the stationary probabilities are then solved by using simple recursive method. Model considered will be discussed in Section 2. The steps for deriving stationary probability equations will be shown in Section 3 and followed by the method to solve these equations in Section 4. Numerical example is shown in Section 5 and in the last section, conclusion remarks will be presented. 2 Model description A single-server queue with RCH killing discipline is considered. Poisson arrival rate of negative customers is denoted as γ. The service time for positive customers is also assumed to have exponential distribution with rate μ. Let f(t) be the probability density function of the interarrival time of positive customers and 퐹̅(푡)is the respective survival function. When t→ 0, 푓(푡)⁄퐹̅(푡) tends to a constant. The service provided to the positive customer is on first come first serve basis. 3 Stationary probabilities A set of equations will be derived in this section. These equations will be used to compute for the stationary probabilities. First, segment the time axis into equal size of length ∆푡. Let 휏푘 = ((푘 − 1)∆푡, 푘∆푡 ) be the kth interval for k = 1, 2, 3, … . Denote k as the hazard rate function and 푓(푘∆푡) = , 1 ≤ 푘 ≤ 퐽 (1) k 퐹̅(푘∆푡) where J is an integer such that 휆퐽 ≃ lim . ∆푡 will be the approximate probability that 푘→∞ given no positive customer arrives in 휏1, 휏2, ⋯ , 휏푘−1 , a positive customer arrives in the interval 휏푘. 2 ITM Web of Conferences 36, 04001 (2021) https://doi.org/10.1051/itmconf/20213604001 ICMSA 2020 Suppose that at time t = 0, a positive customer arrives and the service provided starts immediately. Let M k be the state number of the service process at the end of the interval 휏푘. is defined as follows: 0, if there is no customer in the system; or a negative arrival in k , k 1; or M k = a completion of service occurs in k , k 1. (2) 1, if no completion of service occurs in k , k 1. Next, let k and k be the state numbers for the arrivals of the positive and negative customers, respectively. Then, 0, if k = 0; or kk = a positive arrival in , k 1. (3) min(k , J) , if no positive arrival in k , k 1. 0, if k = 0; or kk = no negative arrival in , k 1. (4) 1, if a negative arrival in k , k 1. Denote mk as the queue length at the end of the interval k . Let Zk=( m k,,, k k M k ) be the state vector at the end of 휏푘 that the queue length is 푚푘 and - the state of the positive arrival is k ; - the state of the negative arrival is k ; and - the state of the service processes is . Suppose Zkk−1 =( m −1, j − 1, 0,1) , then the list of possible events that could occur at the end of 휏푘 with the resulting Z k is as follows: 1. An arrival of positive customer to the system and Zmkk=+( 1, 0, 0,1) . 2. An arrival of negative customer to the system and Zkk=−( m1, j *,1, 0) . 3. A completion of service occurs and Zkk=−( m1, j *, 0, 0) . 4. None of the above three events occurs, hence Zkk= ( m, j *, 0,1) . where j*= min( j , J ) . If mk = 0 at the end of k −1 , then the corresponding state vector is Zjk−1 =−(0, 1, 0, 0) . Hence, the list of possible events with the resulting is: 1. An arrival of positive customer to the system and Zk = (1, 0, 0, 0) . 2. An arrival of negative customer to the system and Zjk = (0, *,1, 0) . 3. No positive or negative arrivals and Zjk = (0, *, 0, 0) . (k ) Let the probability at the end of 휏푘 be pmjrk and the state vector 푍푘 = (푚, 푗, 푟, 푘). Assume (푘) that 푝푚푗푟푘 = lim 푝 exists. If 푍푘−1 = (2, 푗 − 1, 0, 1) and a positive customer arrives 푘→∞ 푚푗푟푘 in 휏푘, we will get 3 ITM Web of Conferences 36, 04001 (2021) https://doi.org/10.1051/itmconf/20213604001 ICMSA 2020 p(kk) = p( −1) t . 3001 2( j− 1) 01 ( j ) (5) k → When , p3001 p2( j− 1) 01 ( j t) . Similarly, by considering the possible events that could occur in 휏푘 , the following set of equations will be obtained. When m = 0, p p + p + p t mj00 ( (m+1)( j − 1) 00( m + 1)( j − 1) 01( m + 1)( j − 1) 10 )( ) for 21jJ − , (6) +p + p1 − t − t ( m( j−−1) 00 m( j 1) 10 )( j ) p p + p + p t mJ 00 ( (m+1)( J − 1) 00( m + 1)( J − 1) 01( m + 1)( J − 1) 10 )( ) +p + p + p t , ( (m+1) J 00( m + 1) J 01( m + 1) J 10 )( ) (7) +p + p + p + p1 − t − t ( m( J−−1) 00 m( J 1) 10 mJ00 mJ 10 )( J ) p p + p + p t mj10 ( (m+1)( j − 1) 00( m + 1)( j − 1) 01( m + 1)( j − 1) 10 )( ) for , (8) +p + p t ( m( j−−1) 00 m( j 1) 10 )( ) p p + p + p t mJ 00 ( (m+1)( J − 1) 00( m + 1)( J − 1) 01( m + 1)( J − 1) 10 )( ) +p + p + p t ( (m+1) J 00( m + 1) J 01( m + 1) J 10 )( ) (9) +p + p + p + p t ( m( J−−1) 00 m( J 1) 10 mJ00 mJ 10 )( ) For m 0 , , pm100 p(m+1) 001 t (10) .